Semi-classical analysis for Fractional Schr\"{o}dinger Equations with fast decaying potenials

TL;DR

This paper analyzes fractional Schr"{o}dinger equations with rapidly decaying potentials, demonstrating the existence of solutions concentrating at local minima using penalized techniques, even when the potential decays arbitrarily or is compactly supported.

Contribution

It introduces a novel approach to find solutions concentrating at local minima for fractional Schr"{o}dinger equations with highly decaying potentials, expanding the class of potentials considered.

Findings

Solutions concentrate at local minima of V(x).

Existence of solutions for a broad class of decaying potentials.

Applicable to potentials with arbitrary decay or compact support.

Abstract

We study the following fractional Schr\"{o}dinger equation \begin{equation*}\label{eq0.1} \epsilon^{2s}(-\Delta)^s u + V(x)u = |u|^{p - 2}u, \,\,x\in\,\,\mathbb{R}^N, \end{equation*} where $s\in (0,\,1)$, $N>2s$, $p>1$ is subcritical and $V(x)$ is a nonnegative continuous potential. We use penalized technique to show that the problem has a family of solutions concentrating at a positive local minimum of $V(x)$ provided that $\frac{2s}{N-2s}+2<p<\frac{2N}{N-2s}$. The novelty is that $V$ can decay arbitrarily or even be compactly supported.

Equations

$${ϵ}^{2s}(-\Delta {)}^{s}u+V(x)u=|u{|}^{p-2}u,x\in {\mathbb{R}}^N,$$

$$iϵ\frac{\partial \psi }{\partial t}={ϵ}^{2s}(-\Delta {)}^s\psi +U(x)\psi -f(\psi )x\in {\mathbb{R}}^N.$$

$$\underset{|x|\to \infty }{\mathrm{liminf}}V(x)|x{|}^{2s}=0.$$

$$H^{s}(\mathbb{R}^{N})=\Big{\{}u\in L^{2}(\mathbb{R}^{N}):\frac{u(x)-u(y)}{|x-y|^{N/2+s}}\in L^{2}(\mathbb{R}^{N}\times\mathbb{R}^{N})\Big{\}},$$

$$\|u\|_{H^{s}(\mathbb{R}^{N})}=\Big{(}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}+u^{2}\,{\mathrm{d}}x\Big{)}^{\frac{1}{2}},$$

$${\int }_{{\mathbb{R}}^N}|(-\Delta {)}^{s/2}u{|}^2\mathrm{d}x={\int }_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y){|}^2}{|x-y{|}^{N+2s}}\mathrm{d}x\mathrm{d}y.$$

$$\Vert u{\Vert }^2={\int }_{{\mathbb{R}}^N}|(-\Delta {)}^{s/2}u{|}^2\mathrm{d}x={\int }_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y){|}^2}{|x-y{|}^{N+2s}}\mathrm{d}x\mathrm{d}y.$$

$$(-\Delta {)}^{s}u(x)$$

$$=C(N,s)\underset{ϵ\to 0}{\lim }{\int }_{{\mathbb{R}}^N\backslash B_{ϵ}(x)}\frac{u(x)-u(y)}{|x-y{|}^{N+2s}}\mathrm{d}y.$$

$$(-\Delta {)}^{s}u(x)={\int }_{{\mathbb{R}}^N}\frac{u(x)-u(y)}{|x-y{|}^{N+2s}}\mathrm{d}y.$$

$${\mathcal{D}}_{V,ϵ}^s({\mathbb{R}}^N)=\left\{u\in {\dot{H}}^s({\mathbb{R}}^N):u\in L^2({\mathbb{R}}^N,V(x)\mathrm{d}x)\right\},$$

$$\|u\|_{\mathcal{D}^{s}_{V,\epsilon}(\mathbb{R}^{N})}=\Big{(}\int_{\mathbb{R}^{N}}\epsilon^{2s}|(-\Delta)^{s/2}u|^{2}+Vu^{2}\,{\mathrm{d}}x\Big{)}^{\frac{1}{2}}.$$

$$0<\lambda =\underset{\Lambda }{\inf }V<\underset{U\backslash \Lambda }{\inf }V.$$

$$\underset{ϵ\to 0}{\lim }V(x_{ϵ})=\underset{\Lambda }{\min }V(x)$$

$$u_{ϵ}(x)\le \frac{C{ϵ}^{\alpha }}{{ϵ}^{\alpha }+|x-x_{ϵ}{|}^{\alpha }},$$

$$I_{ϵ}(u):=\frac{1}{2}{\int }_{{\mathbb{R}}^N}({ϵ}^{2s}|(-\Delta {)}^{s/2}u{|}^2+V|u{|}^2)-\frac{1}{p+1}{\int }_{{\mathbb{R}}^N}|u{|}^p.$$

$${ϵ}^{2s}(-\Delta {)}^s{u}_{ϵ}+V(x)u_{ϵ}={\chi }_{\Lambda }{u}_{ϵ}^{p-1}+{\chi }_{{\mathbb{R}}^N\backslash \Lambda }\min \{P_{ϵ}(x)u_{ϵ},u_{ϵ}^{p-1}\}$$

$${\int }_{{\mathbb{R}}^N}\frac{|u(x){|}^2}{|x{|}^{2s}}\mathrm{d}x\le C_{N,s}|(-\Delta {)}^{s/2}{|}_2^2$$

$$\Vert u{\Vert }_q\le C\Vert (-\Delta {)}^{s/2}u{\Vert }_2^{\theta }\Vert u{\Vert }_2^{1-\theta },$$

$$H^s({\mathbb{R}}^N)\subset \subset L_{loc}^q({\mathbb{R}}^N)\text{compactly, if}q\in [1,2_s^{\ast }).$$

$$\begin{array}{r}{\mathcal{P}}_{ϵ}(x)=0\text{for all}x\in \Lambda \end{array}$$

$$\underset{ϵ\to 0}{\lim }\underset{{\mathbb{R}}^N\backslash \Lambda }{\sup }{\mathcal{P}}_{ϵ}(x){ϵ}^{-(2s+3\kappa /2)}|x{|}^{2s+\kappa }=0,$$

$${\int }_A{\mathcal{P}}_{ϵ}(x)|u{|}^2\le C_{N,s}\frac{{ϵ}^{2s+\frac{3\kappa }{2}}}{{\inf }_{x\in ({\mathbb{R}}^N\backslash \Lambda )\cap A}|x{|}^{\kappa }}{\int }_{{\mathbb{R}}^N}|(-\Delta {)}^{s/2}u{|}^2$$

$${ϵ}^{2s}(-\Delta {)}^{s}u+Vu={\chi }_{\Lambda }(x)u_{+}^{p-1}+{\chi }_{{\mathbb{R}}^N\backslash \Lambda }(x)\min (u_{+}^{p-1},{\mathcal{P}}_{ϵ}(x)u_{+}).$$

$$u_{ϵ}\le {\mathcal{P}}_{ϵ}\text{on}{\mathbb{R}}^N\backslash \Lambda ,$$

$$g_{ϵ}(x,s):={\chi }_{\Lambda }(x)s_{+}^{p-1}+{\chi }_{{\mathbb{R}}^N\backslash \Lambda }(x)\min (s_{+}^{p-1},{\mathcal{P}}_{ϵ}(x)s_{+}),$$

$$G_{ϵ}(x,t)={\int }_0^t{g}_{ϵ}(x,s)\mathrm{d}s,$$

$${\mathbf{g}}_{ϵ}(u)(x)=g_{ϵ}(x,u(x))$$

$${\mathbf{G}}_{ϵ}(u)(x)=G_{ϵ}(x,u(x)).$$

$$\mathbf{J}_{\epsilon}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}\big{(}\epsilon^{2s}|(-\Delta)^{s/2}u|^{2}+V(x)|u|^{2}\big{)}-\int_{\mathbb{R}^{N}}\mathbf{G}_{\epsilon}(u).$$

Full text

[TABLE]

where $s\in(0,\,1)$, $N>2s$, $p>1$ is subcritical and $V(x)$ is a nonnegative continuous potential. We use penalized technique and variational methods to show that the problem has a family of solutions concentrating at a local minimum of $V(x)$ as $\epsilon\to 0$ provided that $\frac{2s}{N-2s}+2<p<\frac{2N}{N-2s}$. The novelty is that $V$ can decay arbitrarily or even be compactly supported.

**Key words: ** fractional Schrödinger; local minimum; compactly supported potential; variational; penalized.

1. Introduction and main results

In this paper, we consider the fractional Schrödinger equation

[TABLE]

where $N>2s$, $s\in(0,1)$, $\epsilon>0$ is a small parameter, $p\in(2,2^{*}_{s})$, $2^{*}_{s}=2N/(N-2s)$, $V(x)\in C(\mathbb{R}^{N},[0,+\infty))$ is a continuous nonnegative potential. Problem (1.1) arise from the study of time-independent waves $\psi(x,t)=e^{-iEt/\epsilon}u(x)$ of the following nonlinear fractional Schrödinger equation

[TABLE]

Equation (1.1) has a wide application in Physics, for example, the Einstein’s theory of relativity, phase transition, conservation laws and fractional quantum mechanic, see [29, 30, 23] and the references therein for more details. The parameter $\epsilon$ is the adimensionalised Planck constant; when $\epsilon$ goes to [math], the quantum physics will shift to the classical physics, in which case the analysis for (1.1) is called semiclassical analysis. In the local case $s=1$, the existence and concentration of solutions for equation (1.1) have been studied extensively, see [2, 3, 14, 15, 19, 20, 31, 32, 4, 22] and their references therein for example.

In the nonlocal case $0<s<1$, to our best knowledge, there are few results on existence and concentration phenomena for (1.1). Actually, due to the nonlocal effect of $(-\Delta)^{s}$, the techniques created for local case can not be applied easily, see [24, 25] for example. Considering the non-vanishing case, i.e., $\inf_{x\in\mathbb{R}^{N}}V(x)>0$, Alves et al. in [1] used the penalized method developed by Del Pino et al in [20] and extension technical developed by Caffarelli et al. in [13] to construct a family of positive solutions that concentrate at a local minimum of $V$ as $\epsilon\to 0$. Also in the non-vanishing case, references [12, 28, 34] proved that the concentration phenomena occurs at the nondegenerate critical points of $V$ by using a reduction method. For the other results of (1.1) with non-vanishing potentials, we would like to refer the readers to [24, 25, 16, 6].

Throughout all the works before, an assumption that $\inf_{\mathbb{R}^{N}}V>0$ was required. The study of the case that $V>0$ but $\inf_{\mathbb{R}^{N}}V=0\,(\liminf_{|x|\to\infty}V(x)=0$) was initiated by X. An et al. in [7], where it was proved by constructing a new penalized function that equation (1.1) has a family of positive solutions concentrating at a positive local minimum of $V$ if $\liminf_{|x|\to\infty}V(x)|x|^{2s}>0$.

In this paper, we try to solve the question of existence and concentration for more general potentials including fast decaying potentials, i.e. potentials for which

[TABLE]

A special case of a fast decaying potential is a potential $V$ that vanishes identically, i.e., $V$ is compactly supported.

In order to state our main result, we need to give some notations and assumptions. For $s\in(0,1)$, the fractional Sobolev space $H^{s}(\mathbb{R}^{N})$ is defined as

[TABLE]

endowed with the norm

[TABLE]

where

[TABLE]

Like the classical case, we define the space $\dot{H}^{s}(\mathbb{R}^{N})$ as the completion of $C^{\infty}_{c}(\mathbb{R}^{N})$ under the norm

[TABLE]

Generally, the fractional Laplacian (see [23] for example) is defined as

[TABLE]

For the sake of simplicity, we define for every $u\in\dot{H}^{s}(\mathbb{R}^{N})$ the fractional $(-\Delta)^{s}$ as

[TABLE]

Our solutions will be found in the following weighted fractional Sobolev space:

[TABLE]

endowed with the norm

[TABLE]

For the potential term $V$, we assume that V\in C\big{(}\mathbb{R}^{N},[0,\infty)\big{)} and

$(\mathcal{A})$ there exist open bounded sets $\Lambda\subset\subset U$ with smooth boundaries $\partial\Lambda,\ \partial U$, such that

[TABLE]

Without loss of generality, we assume that $0\in\Lambda$.

Now, with the notations and assumptions above at hand, we are in a position to state our main result:

Theorem 1.1**.**

Let $N>2s$, $s\in(0,1)$, $p\in(2+\frac{2s}{N-2s},2^{*}_{s})$ and V\in C\big{(}\mathbb{R}^{N},[0,\infty)\big{)} satisfy the assumption $(\mathcal{A})$. Then there exists an $\epsilon_{0}>0$ such that problem (1.1) has a positive solution $u_{\epsilon}\in\mathcal{D}^{s}_{V,\epsilon}(\mathbb{R}^{N})$ if $\epsilon\in(0,\epsilon_{0})$. Moreover, there exist an $\alpha\in(2s/(p-2),N-2s)$ and a family of points $\{x_{\epsilon}\in\overline{\Lambda}:\epsilon\in(0,\epsilon_{0})\}$ such that $u_{\epsilon}(x_{\epsilon})=\sup_{x\in\Lambda}u_{\epsilon}(x)$,

[TABLE]

and

[TABLE]

where $C$ is positive constant.

Particularly, we solve the fractional version Ambrosetti-Malchiodi question([5]) completely, which in the local case $s=1$ have been settled recently (see [9, 10, 11, 35]).

Our approach in this work follows the variational penalization scheme in [7], which was first introduced in [20]. Naturally, equation (1.1) is the Euler-Lagrange equation of the functional

[TABLE]

However, an elementary (but tedious) calculation shows that when $2<p<2^{*}_{s}$ and $V$ vanishes faster than $|x|^{-2s}$, the second integral is not finite in $\mathcal{D}^{s}_{V,\epsilon}(\mathbb{R}^{N})$. del Pino and Felmer provided an effective way to overcome such difficulty: to modify the nonlinear term. According to the idea, assuming that $V$ satisfies $\liminf_{|x|\to\infty}V(x)|x|^{2s}>0$, in [7] the penalized problem

[TABLE]

was considered, where $P_{\epsilon}(x)$ satisfies $P_{\epsilon}(x)=0$ on $\Lambda$, $P_{\epsilon}(x)>0$ outside $\Lambda$ and $P_{\epsilon}(x)\leq\epsilon^{\nu}V(x)$ outside $\Lambda$ with $\nu>0$ is a small parameter. A penalized solution then follows by the fact that the Euler-Lagrange functional corresponding to the penalized problem owns Mountain Pass geometry and is a class of $C^{1}$. One has then to show that $u^{p-2}_{\epsilon}\leq P_{\epsilon}$ outside $\Lambda$. However, when $V$ is compactly supported this approach fails, because one should then have that the solution $u_{\epsilon}$ are compactly supported. In brief, the potential $V$ can not dominated the nonlinear term if it is compactly supported. Our key observation in this paper is that, in order to overcome this difficulty, we can modify the problem by the following fractional Hardy inequality: There exists a positive constant $C_{N,s}$ such that

[TABLE]

for all $u\in\dot{H}^{s}(\mathbb{R}^{N})$(see [26])

One difference between equations in local case $s=1$ and nonlocal case $0<s<1$ is the construction of penalized function. In fact, for a smooth function $f\in C^{\infty}_{c}(\mathbb{R}^{N})$, one can not compute $(-\Delta)^{s}f$ as precisely as $-\Delta f$. Another difference is the energy estimates. The nonlocal effect makes us have to know the global $L^{2}$-norm information of penalized solution, which in [7] was given by the assumption $\liminf_{|x|\to 0}V(x)|x|^{2s}>0$. But, in the present paper, the global $L^{2}$-norm information of penalized solution will be obtained by the Hardy inequality above (see (3.1) in the proof of Lemma 3.3 for more details).

The paper is organized as follows: in Section 2, we establish the penalized scheme and obtain a penalized solution $u_{\epsilon}$. In Section 3 we study the concentration phenomenon of $u_{\epsilon}$. In Section 4, we prove the penalized solution $u_{\epsilon}$ solves the origin problem by constructing a special penalized function. In Section 5, we give a short proof for the existence and concentration for equation (1.1) with more general nonlinear term.

2. The penalized problem

The following inequality exposes the relationship between $H^{s}(\mathbb{R}^{N})$ and the Banach space $L^{q}(\mathbb{R}^{N})$.

Proposition 2.1**.**

(Fractional version of the Gagliardo$-$Nirenberg inequality [23]) For every $u\in H^{s}(\mathbb{R}^{N})$,

[TABLE]

where $q\in[2,2^{*}_{s}]$ and $\theta$ satisfies $\frac{\theta}{2^{*}_{s}}+\frac{(1-\theta)}{2}=\frac{1}{q}$.

By the proposition above, we know that $H^{s}(\mathbb{R}^{N})$ is continuously embedded into $L^{q}(\mathbb{R}^{N})$ for $\ q\in[2,2^{*}_{s}]$. Moreover, on bounded set, the embedding is compact (see [23]), i.e.,

[TABLE]

Now we are going to modify the origin problem (1.1). According to the fractional Hardy inequality (1.4), we choose a family of penalized potentials $\mathcal{P}_{\epsilon}\in L^{\infty}(\mathbb{R}^{N},[0,\infty))$ for $\epsilon>0$ small in such a way that

[TABLE]

and

[TABLE]

where $\kappa>0$ is a small parameter. Noting that by (1.4), when $\epsilon>0$ is small enough, it holds

[TABLE]

where $C_{N,s}$ is the constant in (1.4). This type of estimate plays a key role in the paper. Now we give the penalized problem according to the choice of $\mathcal{P}_{\epsilon}$:

[TABLE]

It is easy to check that if a solution $u_{\epsilon}$ of (2.4) satisfies

[TABLE]

then $u_{\epsilon}$ is a solution of (1.1).

For simplicity, we define

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Then the Euler-Lagrange functional of (2.4) is

[TABLE]

By the Hardy inequality (1.4), we know that $\mathbf{J}_{\epsilon}$ is well-defined on $\mathcal{D}^{s}_{V,\epsilon}(\mathbb{R}^{N})$. In order to obtain a critical point to $\mathbf{J}_{\epsilon}$ via variational method, we need to verify that $\mathbf{J}_{\epsilon}$ is $C^{1}$ and satisfies the (P.S.) condition.

Lemma 2.2**.**

$\mathbf{J}_{\epsilon}:\mathcal{D}^{s}_{V,\epsilon}(\mathbb{R}^{N})\to\mathbb{R}$* is $C^{1}$ and satisfies (P.S.) condition.*

Proof.

We only need to show that the nonlinear term

[TABLE]

is $C^{1}$ since the other terms are obvious. For every $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N})$ and $0<|t|<1$, we have

[TABLE]

The existence of first order Gateaux derivative then follows by Dominated Convergence Theorems.

Let $(u_{n}),\ u\in\mathcal{D}^{s}_{V,\epsilon}(\mathbb{R}^{N})$ satisfy $u_{n}\to u\in\mathcal{D}^{s}_{V,\epsilon}(\mathbb{R}^{N})$. By Proposition 2.1, the construction of $\mathcal{P}_{\epsilon}$, (1.4) and Dominated Convergence Theorem, we deduce that for every $\varphi\in H^{s}_{V,\epsilon}(\mathbb{\mathbb{R}^{N}})$ with $\|\varphi\|_{H^{s}_{V,\epsilon}(\mathbb{R}^{N})}\leq 1$,

[TABLE]

which says that $\mathbf{J}_{\epsilon}$ is a class of $C^{1}$.

Next we show that $\mathbf{J}_{\epsilon}$ satisfies (P.S.) condition, i.e., to prove that any sequence $\{u_{n}\}\subset\mathcal{D}^{s}_{V,\epsilon}(\mathbb{R}^{N})$ satisfying

[TABLE]

is relatively compact.

It is standard to verify using the fact $p>2$ and the construction of $\mathcal{P}_{\epsilon}$ that $\{u_{n}\}$ is bounded in $\mathcal{D}^{s}_{V,\epsilon}(\mathbb{R}^{N})$. By Proposition 2.1, one has thus $u_{n}\to u$ in $L^{p}_{loc}(\mathbb{R}^{N})$.

On the other hand, one has, for every $\sigma>0$, by the fractional Hardy inequality (1.4), for $R>0$ large enough,

[TABLE]

Hence

[TABLE]

Since $\sigma>0$ is arbitrary, this proves the lemma.

∎

It is standard to show that $\mathbf{J}_{\epsilon}$ owns Mountain Pass geometry. Hence, from Lemma 2.2, we can use the min-max procedure in [37] to find a critical point for the functional $\mathbf{J}_{\epsilon}$.

Lemma 2.3**.**

The Mountain Pass value

[TABLE]

*can be achieved by a positive function $u_{\epsilon}$ which is $C^{1,\gamma}$ for some $\gamma\in(0,1)$ and is a solution to the penalized problem (2.4), where *

[TABLE]

Proof.

The existence of $u_{\epsilon}$ can be checked by the Mountain Pass Lemma in [37]. By the regularity argument in [25, Appendix D], $u_{\epsilon}$ is $C^{1,\gamma}$ for some $\gamma\in(0,1)$. Testing the penalized equation (2.4) with $(u_{\epsilon})_{-}$ and integrating, we can see that $u_{\epsilon}\geq 0$.

Supposing to the contrary that there exists $x_{0}\in\mathbb{R}^{N}$ such that $u_{\epsilon}(x_{0})=0$, we then have

[TABLE]

which is a contradiction. Therefore, $u_{\epsilon}>0$.

∎

3. Concentration

In this section, we prove that the solution $u_{\epsilon}$ obtained by Lemma 2.3 will concentrate at a local minimum of $\Lambda$ as $\epsilon\to 0$ via comparing the energy $c_{\epsilon}$ in Lemma 2.3 with the least energy of the limiting problem corresponding to (1.1). This concentration can help to linearize the penalized equation (2.4) (see (4.2) below).

3.1. The limiting problem

For $a>0$, the limiting problem associated to (1.1) is

[TABLE]

whose Lagrange-Euler functional $J_{a}:H^{s}(\mathbb{R}^{N})\to\mathbb{R}$ is defined as

[TABLE]

From [25, 27], we know that the limiting problem has a positive ground state. We define the limiting energy by

[TABLE]

Since for every $v\in H^{s}(\mathbb{R}^{N}),\ J_{a}(|v|)\leq J_{a}(v)$, by the density of $C_{0}^{\infty}(\mathbb{R}^{N})$ in $H^{s}(\mathbb{R}^{N})$, we have

[TABLE]

The following proposition is obvious.

Proposition 3.1**.**

Let $a>0$ and $v\in H^{s}(\mathbb{R}^{N})$. Define

[TABLE]

Then

[TABLE]

In particular, $v$ is a solution of $(\mathcal{P}_{1})$ if and only if $v_{a}$ is a solution of $(\mathcal{P}_{a})$ and

[TABLE]

Moreover, $\mathcal{C}({a})$ is continuous and increasing in $(0,\infty)$.

Now we are going to compare the energy $c_{\epsilon}$ with $\mathcal{C}(V(x)),x\in\Lambda$. Firstly, we give the upper bound of the energy $c_{\epsilon}$:

Lemma 3.2**.**

(Upper bound of the energy) we have

[TABLE]

Proof.

For any given $x_{0}\in\Lambda$ and nonnegative function $v\in C^{\infty}_{c}(\mathbb{R}^{N})\backslash\{0\}$, we define

[TABLE]

Obviously, $supp\,\,v\subset\Lambda$ when $\epsilon$ is small enough. It is easy to check that $\gamma_{\epsilon}(t)=tT_{0}v_{\epsilon}\in\Gamma_{\epsilon}$ for some $T_{0}$ large enough. So

[TABLE]

and

[TABLE]

which implies our conclusion. ∎

Secondly, we give the lower estimate on energy $c_{\epsilon}$.

Lemma 3.3**.**

Let $\{\epsilon_{n}\}_{n\in\mathbb{N}}$ be a sequence of positive numbers converging to [math], $\{u_{n}\}$ be a sequence of critical points given by Lemma 2.3, and for $j\in{1,2,\cdots,k}$, $\{x^{j}_{n}\}$ be a sequence in $\mathbb{R}^{N}$ converging to $x^{j}_{*}\in\mathbb{R}^{N}$. If

[TABLE]

[TABLE]

and for some $\rho>0$,

[TABLE]

then $x^{j}_{*}\in\bar{\Lambda}$ and

[TABLE]

Some details in the proof of this lemma will be omitted since they are similar to Proposition 3.4 in [7], and more attentions will be paid on the difference caused by the fast decaying potential $V$ (see (3.1) below for more details).

Proof.

For $j\in\{1,\ldots,k\}$, the rescaling function $v^{j}_{n}\in H^{s}_{loc}(\mathbb{R}^{N})$ defined by

[TABLE]

satisfies weakly the rescaled equation

[TABLE]

where $V^{j}_{n}(y)=V(x^{j}_{n}+\epsilon_{n}y),\ \textsf{g}^{j}_{n}(v^{j}_{n})=g_{\epsilon_{n}}(x^{j}_{n}+\epsilon_{n}y,v^{j}_{n}).$ By the assumption,

[TABLE]

the sequence $\{v^{j}_{n}\}_{n\geq 1}$ is bounded in $H^{s}_{loc}(\mathbb{R}^{N})$. By the regularity assertion in Appendix D of [25], the Liouville-type Lemma in Lemma 3.3 of [7], we can conclude that there exist $x^{j}_{*}\in\bar{\Lambda}$ and a nonnegative $v^{j}_{*}\in H^{s}(\mathbb{R}^{N})\backslash\{0\}$ such that $v^{j}_{n}\to v^{j}_{*}$ in $H^{s}_{loc}(\mathbb{R}^{N})\cap C^{1}_{loc}(\mathbb{R}^{N})$, $x^{j}_{n}\to x^{j}_{*}$ and

[TABLE]

Since $v^{j}_{*}\geq 0$, we see

[TABLE]

where $C>0$ is a universal constant.

In order to study the integral outside $B_{\epsilon_{n}R}(x^{j}_{n})$, we choose cut-off function $\eta(x)\in C^{\infty}(\mathbb{R}^{N})$ such that $0\leq\eta\leq 1,\ \eta=0$ if $0\leq|x|\leq 1$ and $\eta=1$ when $|x|\geq 2$. Define

[TABLE]

Since $u_{n}$ is a solution to the penalized problem $\eqref{eq2.2}$, we have by taking $\psi_{n,R}u_{n}$ as a test function in the penalized problem $\eqref{eq2.2}$ that

[TABLE]

Hence

[TABLE]

By scaling, $B_{\epsilon_{n}R}(x^{j}_{n})\cap B_{\epsilon_{n}R}(x^{l}_{n})=\emptyset$ if $n$ is large enough. Hence, by the fact that $v^{j}_{n}\to v^{j}_{*}$ in $L^{p}_{loc}(\mathbb{R}^{N})$, we have

[TABLE]

Now we estimate $R_{n}$. A change of variable tells us

[TABLE]

where the functions $\beta^{l}_{n}$ and $\alpha^{l}_{n}$ are defined skillfully as

[TABLE]

and

[TABLE]

Following, we have

[TABLE]

By the choice of $\eta$ and $\lim\limits_{n\to\infty}\frac{|x^{l}_{n}-x^{s}_{n}|}{\epsilon_{n}}=\infty$ if $l\neq s$, for $n$ large, we have

[TABLE]

and

[TABLE]

Also, for large $n$,

[TABLE]

For $|R^{(i2)}_{n}|,\ i=1,2$, it holds

[TABLE]

By the fractional Hardy inequality (1.4) and letting $\widetilde{R}=R^{\frac{N+1}{N}}$, we find

[TABLE]

For $R^{(11)}_{n}$, by the estimates of $R^{(i2)}_{n}$ above and the similar estimates of Proposition 3.4 in [7], we have

[TABLE]

Similarly, we get

[TABLE]

and

[TABLE]

Therefore

[TABLE]

Finally, letting $R\to\infty$, we have

[TABLE]

∎

At last, as a result of Lemma 3.3, we have the following concentration for the solution $u_{\epsilon}$.

Lemma 3.4**.**

Let $\rho>0$. There exists a family of points $\{x_{\epsilon}\}\subset\Lambda$ such that

[TABLE]

Proof.

It is easy to verify using the construction of $\mathcal{P}_{\epsilon}$ that

[TABLE]

Then by the regularity assertion in Appendix D of [25], we get the existence of $x_{\epsilon}\in\bar{\Lambda}$. We assume that $x_{\epsilon}\to x_{*}$.

By Lemma 3.3, it holds

[TABLE]

But,

[TABLE]

Hence by the monotonicity of $\mathcal{C}(\cdot)$, $V(x_{*})=\inf_{\Lambda}V(x).$

Arguing by contradiction, if $(iii)$ is not hold, we can infer from Lemma 3.3 that

[TABLE]

which is a contradiction. The proof is then completed.

∎

4. Back to the original problem

In this section, we show that $u_{\epsilon}^{p-2}\leq\mathcal{P}_{\epsilon}$ by comparison principle.

By Lemma 3.4 and the $L^{\infty}$ estimate in [25, Appendix D], we can assume that there exists a positive constant $C_{\infty}$ such that

[TABLE]

if $\epsilon$ is small enough, from which we can linearize the penalized problem (2.4) outside small balls as follows.

Proposition 4.1**.**

For $\epsilon>0$ small enough and $\delta\in(0,1)$, there exist $R>0$ and $x_{\epsilon}\in\Lambda$ such that

[TABLE]

Letting $v_{\epsilon}(x)=u_{\epsilon}(\epsilon x+x_{\epsilon})$, it is easy to check that

[TABLE]

where $V_{\epsilon}(\cdot)=u_{\epsilon}(\epsilon\cdot+x_{\epsilon})$ and $\widetilde{P}_{\epsilon}(\cdot)=\mathcal{P}_{\epsilon}(\epsilon\cdot+x_{\epsilon})$.

Now we construct a suitable sup-solution to equation (4.3). Let ${\tilde{\eta}_{\beta}}(|x|):\mathbb{R}^{N}\to[0,1]$ be a smooth non-increasing function with ${\tilde{\eta}_{\beta}}(s)\equiv 1$ when $s\in[-1,1]$ and $\tilde{\eta}_{\beta}\equiv 0$ when $s\in(-\infty,-1-\beta)\bigcup(1+\beta,+\infty)$, where $\beta>0$ is a small parameter. Define $\eta_{\beta}(|x|)=\tilde{\eta}_{\beta}(R|x|)$. It is worth mentioning that the non-increasing of $\eta$ is important, see (4) below. Setting $0<\alpha<N-2s$ and denoting

[TABLE]

we have:

Proposition 4.2**.**

Let $\epsilon>0$ be small enough. Then for every $x\in\mathbb{R}^{N}\backslash B_{R}(0)$, it holds

[TABLE]

Proof.

For $x\in\mathbb{R}^{N}\backslash B_{2R}(0)$, letting $\beta>0$ be small enough, it holds

[TABLE]

When $x\in B_{2R}(0)\backslash B_{R}(0)$, by the construction of $\eta_{\beta}$ and the computation above, it holds

[TABLE]

where $C_{\beta}$ is a positive constant depending only on $\beta$.

For $K^{\beta}_{\alpha}$, by Change-Of-Variable Theorem and the non-increasing of $\tilde{\eta}_{\beta}$, it holds

[TABLE]

where $\widetilde{C}_{\alpha}$ is a constant depending only on $\alpha$(it will hold $\widetilde{C}_{N-2s}>0$ if $\alpha=N-2s$).

Finally, by the computation above, we conclude that if $R$ is large enough and $\epsilon$ is small enough, it holds

[TABLE]

for every $x\in\mathbb{R}^{N}\backslash B_{R}(0)$, where $\Lambda_{\epsilon}=\{x:\epsilon x+x_{\epsilon}\in\Lambda\}$. This completes the proof.

∎

At the last of this section, we give the proof of Theorem 1.1

Proof of Theorem 1.1. Let

[TABLE]

where the function $f^{\beta}_{\alpha}$ is that in Proposition 4.2.

It is easy to check that $\mathcal{P}_{\epsilon}$ satisfies the assumption (2.2).

By Proposition 4.2, letting the constant $C>0$ above be large enough and $\tilde{v}_{\epsilon}(x)=\overline{U}(x)-v_{\epsilon}(x)$, we have

[TABLE]

Then, since $\tilde{v}_{\epsilon}\in\dot{H}^{s}(\mathbb{R}^{N})$ (when $\alpha$ is closed to $N-2s$), testing the equation above against with $\tilde{v}^{-}_{\epsilon}(x)$, by the fractional Hardy inequality in (1.4), we find $\tilde{v}^{-}_{\epsilon}(x)=0,\ x\in\mathbb{R}^{N}$. Hence $\tilde{v}_{\epsilon}(x)\geq 0,\ x\in\mathbb{R}^{N}$. Especially, we have

[TABLE]

Moreover, it holds

[TABLE]

As a result, $u_{\epsilon}$ solves the origin problem. This completes the proof.

5. Further results

In this section, we will consider (1.1) with general nonlinearity, i.e.,

[TABLE]

where the potential $V(x)$ is the same as before, the nonlinearity $f:\mathbb{R}\to\mathbb{R}$ is assumed to satisfy the following properties:

$(\textbf{f}_{1})$ $f$ is an odd function and $f(t)=o(t^{1+\tilde{\kappa}})$ as $t\to 0^{+}$, where $\tilde{\kappa}=\frac{2s+2\kappa}{\alpha-\nu}>0$ with $\nu>0$ is a small parameter and $\kappa$ is the parameter in (4.11).

$(\bf{f}_{2})$ $\lim_{t\to\infty}\frac{f(t)}{t^{p}}=0$ for some $1<p<2^{*}_{s}-1$.

$({\bf{f}}_{3})$ There exists $2<\theta\leq p+1$ such that

[TABLE]

where $F(t)=\displaystyle\int_{0}^{t}f(\alpha)\,{\mathrm{d}}\alpha$.

$(\textbf{f}_{4})$ The map $t\mapsto\frac{f(t)}{t}$ is increasing on $(0,+\infty)$.

We have the following results which are same as Theorem 1.1.

Theorem 5.1**.**

Let $N>2s$, $s\in(0,1)$, p\in\Big{(}2+\frac{2s}{N-2s},2^{*}_{s}\Big{)}, $f$ satisfy $(\bf{f}_{1})$- $(\bf{f}_{4})$ and $V$ be the same as before. Then problem (1.1) has a positive solution $\hat{u}_{\epsilon}\in\mathcal{D}^{s}_{V,\epsilon}(\mathbb{R}^{N})$ if $\epsilon>0$ is small enough. Moreover, there exist a $\hat{x}_{\epsilon}\in\Lambda$ and an $\alpha\in(2s/(p-2),2^{*}_{s})$ such that

[TABLE]

and

[TABLE]

where $C$ is positive constant.

Proof.

We define the penalized nonlinearity $\hat{g}_{\epsilon}:\mathbb{R}^{N}\times\mathbb{R}$ as

[TABLE]

In the sequel, we denote $\widehat{G}_{\epsilon}(x,t)=\displaystyle\int_{0}^{t}\hat{g}_{\epsilon}(x,s)\,{\mathrm{d}}s$ and define the penalized superposition operators $\hat{\mathbf{g}}_{\epsilon}$ and $\widehat{\mathbf{G}}_{\epsilon}$ as

[TABLE]

Accordingly, the penalized functional $\hat{J}_{\epsilon}:\mathcal{D}^{s}_{V,\epsilon}(\mathbb{R}^{N})\to\mathbb{R}$ is given by

[TABLE]

By conditions $(f_{2})$ and $(f_{3})$, we can verify using the same argument in the proof of Lemma 2.2 that $\hat{J}_{\epsilon}$ is $C^{1}$ and satisfies (P.S.) condition. Then similar to Lemma 2.3, one can show that the Mountain Pass value

[TABLE]

can be achieved by a positive function $\hat{u}_{\epsilon}\in\mathcal{D}^{s}_{V,\epsilon}$, where

[TABLE]

Furthermore, it holds

[TABLE]

From [33, 27] and the condition on $f$, the following limiting problem

[TABLE]

has a positive ground state $\hat{u}$. Moreover, the least energy

[TABLE]

is continuous and increasing. Then, by the same argument in Lemma 3.4, there exists a $\hat{x}_{\epsilon}\in\Lambda$ such that

[TABLE]

As a result, using $(f_{1})$ to linearize (5.2), we get

[TABLE]

Then, by the same argument in Section 4, it holds

[TABLE]

where $\alpha$ is the same as that in (4.11). Following, for every $x\in\mathbb{R}^{N}\backslash\Lambda$, since $\alpha\tilde{\kappa}>2s+2\kappa$, letting $\epsilon>0$ be small enough, it holds

[TABLE]

As a result, $\hat{u}_{\epsilon}$ is a solution to the origin problem (5.1). This completes the paper.

∎

**Acknowledgements: ** L. Duan is partially supported by China Scholarship Council (No.201906770024). Y. Peng is supported by the grant NSFC (No. 11501143).